Not sure it would have been possible to keep pioneers and frontiersmen out of Kentucky, Tennessee and Ohio. Daniel Boone had already blazed the trail by 1778, and by 1790, Kentucky had almost 100,000 people living there. So even if the territory to the West of the mountains is off limits to Americans, they are going to emigrate anyway.
However, if the territory can't be incorporated into the US/13 colonies, then I think the hemmed in country stagnates over the long term. Most immigrants that came to America in the late 18th and early 19th century came for the land and settled in rural areas, with many going West soon after arriving. So the country would have far fewer immigrants. Plus, the high birth rates were tied to a rural/farm culture where children were needed to work the farm, so I think a hemmed in America also has lower birthrates as well. While the cities of the Coast probably continue to grow and industry develops faster, the economy as a whole is much smaller and more stagnant. Slavery cannot prosper either as cotton and tobacco crops wear out the soil and so neither crop will do well after a few seasons, and with no new land to move to, the plantations lose, and slavery will lose its efficiency - and die a slow death.
If France keeps Louisiana, I think you could see lots of immigrants from America and Europe going to settle along the river valleys. Native Americans would fare much better, probably. And somebody eventually settles California. Is it the Spanish, or some other people.